This article is invalid. Nakamoto consensus implies relative immutability, that is a block N deep in the chain is more immutable than that of block N+1 in the chain. In the proof of stake, the immutability is corrupted trivially, by all stake participants to sign-of multiple histories, or in some cases, to claim that all possible histories have in fact happened. PoS does not prevent this, and thus double spend can, and will occur.